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1999 Fiscal Year Final Research Report Summary

On the Riemann zeta-function and the 3D hyperbolic variety

Research Project

Project/Area Number 09640068
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNihon University

Principal Investigator

MOTOHASHI Yoichi  Nihon University, College of Science of Technology, Professor, 理工学部, 教授 (30059969)

Project Period (FY) 1997 – 1999
KeywordsZeta-Function / Hyperbolic Space / Unitary Representation
Research Abstract

Our research was a continuation of our former project (grant-in-aid no.07640072). The latter had been devoted to the investigation of a relation between the Riemann zeta-function and automorphic forms of one complex variable, i.e., functions on the upper half-plane which are automorphic with respect to certain arithmetically defined fuchsian groups, especially the modular group and its Hecke congruence subgroups. The aim of the project under summarizing was to extend such a relation to a wider class of zeta-functions and automorphlic functions living in higher dimensional varieties which are, respectively, natural generalizations of the Riemann zeta-function and the hyperbolic upper half-plane. As a first step of our grand project, we took up, in this project, the case of the Dedekind zeta-function of the Gaussian number field. This choice was made not only for the sake of structural simplicity but because of the fact that it raises a genuine new situation much different from that we experienced in our former project. More precisely, we found, with the co-operation of Prof. Bruggeman of Utrecht University, that in the quest of such an extension we need to move to the Lie group SL (2, C) and its unitary representations. In that way we have established an extension of Kuznetsov's sum formula to the complex situation, and, as a consequence, a spectral decomposition of the fourth power moment of the Dedekind zeta-function of the Gaussian number field. These accomplishments fulfill the aim of our project.

  • Research Products

    (17 results)

All Other

All Publications (17 results)

  • [Publications] Y.Motohashi: "The means square of Dedekimd zera-functions of quadratic number fields"Analytic Number Theory (Cambridge University Press). 309-324 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Motohashi: "Trace formula over the hyperbolic upper half space"Analytic Number Theory (Carrbridge University Press). 265-286 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Motohashi: "A note on mean value of the zeta and L-functions. IX"Proc.Japan.Acad (日本学士院紀要). 75. 147-149 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Motohashi: "New analytic problems over imaginary quadratic number fields"Inkeri Memorial Volume. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Motohashi: "A note on the zero-free region of the Riemann zeta-function"A.Sarkozy Festschrift. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Iric, M.Jutila and Y.Notshashi: "The Mcllin transform of the Ricmann zera-function"Acta Anthmetica. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 本橋洋一: "マンゼータ函数と保型波動"共立出版. 201 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Motohashi: "An asymptotic expansion of the square of the Riemann zeta-function"Sieve Methods, Exponential Sums, and their Applications in Number Theory, Cambridge Univ.Press, Cambridge. 293-307 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] The mean square of Dedekind zeta-functions of quadratic number fields.: "ibid"309-324

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Trace formula over the hyperbolic upper half space.: "Analytic Number Theory, Cambridge Univ. Press, Cambridge"265-286 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] The last 50 years in Analytic Number Theory-a personal account-(Japanese): "Sugaku"49. 96-100 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A note on mean value of the zeta and L-functions.: "IX.Proc. Japan Acad."75. 147-149 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Ivic, M.Jutila and Y.Motohashi: "The Mellin transform of the Riemann zeta-function."(Acta Arith., accepted for publication).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Motohashi: "New analytic problems over imaginary quadratic number fields."(Inkeri Memorial Vol., accepted for publication).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Motohashi: "A note on the zero-free region of the Riemann zeta-function."(Sarkozy Festschrift., accepted for publication).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] R.W.Bruggeman and Y.Motohashi: "The sum formula for Kloosterman sums and the fourth moment of the Dedekind zeta-function over the Gaussian number field"(submitted to Acta Mathematica).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Motohashi: "The Riemann zeta-function and automorphic waves.(Japanese)"Kyoritu Shyuppan, Tokyo. (1999)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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